A227898 Number of primes p < n with p + 6 and n + (n - p)^2 both prime.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 4, 3, 2, 2, 3, 3, 3, 3, 3, 4, 4, 2, 2, 3, 3, 3, 2, 2, 5, 5, 2, 5, 4, 2, 4, 5, 2, 7, 5, 3, 4, 5, 3, 3, 4, 4, 3, 5, 4, 9, 9, 2, 5, 3, 4, 8, 6, 2, 5, 8, 3, 4, 7, 3, 10, 5, 2, 7, 4, 5, 10, 6, 4, 6, 6, 2, 6, 8, 3, 6, 5, 3, 6, 6, 5, 9, 4, 5, 7, 5, 4, 9, 10

Offset

1,12

Comments

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) For any integer n > 5, there is a prime p with p + 6 and n*(n - p) - 1 both prime.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Example

a(6) = 1 since 5, 5 + 6 = 11 and 6 + (6 - 5)^2 = 7 are all prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[n+(n-Prime[i])^2], 1, 0], {i, 1, PrimePi[n-1]}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A023201, A185636, A204065, A227899, A230261.

Sequence in context: A025834 A257379 A336456 * A035649 A278509 A094782

Adjacent sequences: A227895 A227896 A227897 * A227899 A227900 A227901

Keyword

nonn

Author

Zhi-Wei Sun, Oct 14 2013

Status

approved